Metric, Curvature, Connection

Question

I am new of riemannian geometry and I am not sure of implications of metrics. Metrics implies riemann connection with no torsion and metric compatibility. Connection implies curvature of the connection or does it only depens on the metric?

Answer 1

Curvature depends on connection. In imprecise words, curvature is the composition of connection and derivation. In general connection is an object associated to vector bundle over a smooth manifold and so is the curvature and hence it has nothing to do with a metric. In case of Riemannian manifold, we have a special connection, namely the Rimennian connection whose existence and uniqueness depends on the Riemannin metric and hence the curvature associated to it also depends on the metric. For more details see Chapter 12 of this lecture note.